The Relation Between Classical and Quantum Mechanics

TL;DR

This paper develops a general theory linking classical and quantum mechanics through pure states and probability, exploring their relationship, measurement, and space-time constraints, and questions traditional notions of classical limits and reduction.

Contribution

It introduces a unified framework based on pure states and quantum logic, providing new insights into the relationship and reduction between classical and quantum mechanics.

Findings

Classical and quantum mechanics can be derived from a common general theory.

Localisation in quantum mechanics relates to compactness and topological states.

Classical mechanics can be seen as weakly reducible to quantum mechanics within certain domains.

Abstract

Familiar formulations of classical and quantum mechanics are shown to follow from a general theory of mechanics based on pure states with an intrinsic probability structure. This theory is developed to the stage where theorems from quantum logic enable expression of the state geometry in Hilbert space. Quantum and classical mechanics are then elaborated and applied to subsystems and the measurement process. Consideration is also given to space-time geometry and the constraints this places on the dynamics. Physics and Mathematics, it is argued, are growing apart; the inadequate treatment of approximations in general and localisation in quantum mechanics in particular are seen as contributing factors. In the description of systems, the link between localisation and lack of knowledge shows that quantum mechanics should reflect the domain of applicability. Restricting the class of states provides a means of achieving this goal. Localisation is then shown to have a mathematical expression in terms of compactness, which in turn is applied to yield a topological theory of bound and scattering states. Finally, the thesis questions the validity of classical limits and quantisations in intertheoretic reduction, and demonstrates that a widely accepted classical limit does not constitute a proof of reduction. It proposes a procedure for determining whether classical and quantum mechanics are weakly equivalent over a domain of applicability, and concludes that, in this restricted sense, classical mechanics reduces to quantum mechanics.